Preheat oven to 425 degrees F. Place the pitas onto a rimmed baking sheet.
Squeeze the cloves of roasted garlic out into a small bowl. Mash the cloves with a fork and stir in 1/4 cup of olive oil. Stir with a spoon to combine. Spoon the garlic and olive oil mixture evenly over the pitas, and use the back of the spoon to spread everything around.
Crumble to goat cheese and sprinkle it evenly over the pitas.
In a small bowl, toss the Brussels sprouts with vinegar, 1 Tablespoon olive oil, salt, and pepper. Spoon the mixture evenly over the pitas.
Sprinkle each pita with the chopped prosciutto. Bake for 8-10 minutes, or until the pitas are golden and crisp around the edges, and the Brussels sprouts are tender and slightly browned.